Crazy crystals

By Paul Steinhardt

YOU HAVE just moved into a new house, and you are choosing tiles to lay on the kitchen floor. Like most people, you don’t want an untidy mixture of shapes, but a single tile shape that repeats again and again. Triangles, rectangles or even hexagons will do&colon; any one of these shapes can be used to cover the floor in a repeating pattern. But there are some shapes that you will not be able to use. Take pentagons, say, and no matter how hard you try to fit them together they will always leave annoying gaps.

The same phenomenon crops up when you start to think about how atoms are arranged in three dimensions in a solid. Most solids fall into one of two classes&colon; crystals and glasses. In a glass, the atoms are arranged completely randomly, so the problem of regular patterns does not arise. But the atoms of a crystal sit in neat, periodic structures, just like three-dimensional kitchen tiles. It’s easy enough to find crystals built around threefold, fourfold or sixfold symmetry-with the atoms arranged in the 3D equivalents of triangles, squares and hexagons. But you will never find a crystal based on fivefold symmetry.

This notion is embodied in a set of mathematical rules that were developed nearly 150 years ago by the French physicist Auguste Bravais, and has long been firmly fixed in the minds of physicists. So people were stunned when Dan Shechtman, now at the Technion in Israel, and his colleagues at the National Bureau of Standards in Gaithersburg, Maryland announced just over a decade ago that they had found …

To continue reading this premium article, subscribe for unlimited access.